In this section, we will derive the Fourier dual of the
Constant OverLap-Add (COLA) condition for STFT analysis windows
(discussed in §7.1). Recall that for perfect reconstruction
using a hop-size of
samples, the window must be
. We
will find that the equivalent frequency-domain condition is that the
window transform must have spectral zeros at all frequencies
which are a nonzero multiple of
. Following established
nomenclature for filter banks, we will say that such a window
transform is
.

Poisson Summation Formula

Consider the summation of N complex sinusoids having frequencies
uniformly spaced around the unit circle [264]:

The ``Nyquist(
)'' property for a function
simply means
that
is zero at all nonzero multiples of
(all harmonics
of the frame rate here).

We may also refer to (8.33) as the ``weak COLA constraint''
in the frequency domain. It gives
necessary and sufficient conditions for perfect reconstruction in
overlap-add FFT processors. However, when the short-time spectrum is
being modified, these conditions no longer apply, and a
stronger COLA constraint is preferable.

Above, we used the Poisson Summation Formula to show that the
constant-overlap-add of a window in the time domain is equivalent to
the condition that the window transform have zero-crossings at all
harmonics of the frame rate. In this section, we look briefly at the
dual case: If the window transform is COLA in the frequency
domain, what is the corresponding property of the window in the time
domain? As one should expect, being COLA in the frequency domain
corresponds to having specific uniform zero-crossings in the time
domain.

Bandpass filters that sum to a constant provides an ideal basis for
a graphic equalizer. In such a filter bank, when all the
``sliders'' of the equalizer are set to the same level, the filter
bank reduces to no filtering at all, as desired.

Let
denote the number of (complex) filters in our filter bank,
with pass-bands uniformly distributed around the unit circle. (We will
be using an FFT to implement such a filter bank.) Denote the
frequency response of the ``dc channel'' by
. Then the
constant overlap-add property of the
-channel filter bank can be
expressed as

(9.35)

which means

(9.36)

where
as usual. By the dual
of the Poisson summation formula, we have

(9.37)

where
means that
is zero at all nonzero
integer multiples of
, i.e.,

(9.38)

Thus, using the dual of the PSF, we have found that a good
-channel
equalizer filter bank can be made using bandpass filters which have
zero-crossings at multiples of
samples, because that property
guarantees that the filter bank sums to a constant frequency response
when all channel gains are equal.

The duality introduced in this section is the basis of the
Filter-Bank Summation (FBS) interpretation of the short-time
Fourier transform, and it is precisely the Fourier dual of the
OverLap-Add (OLA) interpretation [9]. The FBS
interpretation of the STFT is the subject of Chapter 9.

Using ``square-root windows''
in the WOLA context, the
valid hop sizes
are identical to those for
in the OLA case.
More generally, given any window
for use in a WOLA system, it
is of interest to determine the hop sizes which yield perfect
reconstruction.

In a weighted overlap-add system, the following windows can be used
to satisfy the constant-overlap-add condition:

For the rectangular window,
, and
(since
is a sinc function which reduces to
when
, and
.

For the Hamming window, the critically sampled window transform
has three nonzero samples (where the rectangular-window transform has
one). Therefore,
has
nonzero samples at critical
sampling. Measuring main-lobe width from zero-crossing to
zero-crossing as usual, we get
radians per sample, or
``6 side lobes'', for the width of
.

The square of a length
-term Blackman-Harris-family window
(where rect is
, Hann is
, etc.) has a main lobe of width
, measured from zero-crossing to zero-crossing in
``side-lobe units'' (
). This is up from
for the
original
-term window.

The width of the main lobe can be used to determine the
hop size in the STFT, as will be discussed further in
Chapter 9.

Note that we need only find the first zero-crossing in the
window transform for any member of the Blackman-Harris window family
(Chapter 3), since nulls at all harmonics of
that frequency will always be present (at multiples of
).